Rayleigh distribution

<h2 id="definition">Definition</h2>
The <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> of the Rayleigh distribution is<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

f
        (
        x
        ;
        σ
        )
        =
        
          
            x
            
              σ
              
                2
              
            
          
        
        
          e
          
            −
            
              x
              
                2
              
            
            
              /
            
            (
            2
            
              σ
              
                2
              
            
            )
          
        
        ,
        
        x
        ≥
        0
        ,
      
    
    {\displaystyle f(x;\sigma )={\frac {x}{\sigma ^{2}}}e^{-x^{2}/(2\sigma ^{2})},\quad x\geq 0,}

where 
 
 
 
 σ
 
 
 {\displaystyle \sigma }
 
 is the <a href="/facts/Scale_parameter/6lxcyKUf">scale parameter</a> of the distribution. The <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> is<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

F
        (
        x
        ;
        σ
        )
        =
        1
        −
        
          e
          
            −
            
              x
              
                2
              
            
            
              /
            
            (
            2
            
              σ
              
                2
              
            
            )
          
        
      
    
    {\displaystyle F(x;\sigma )=1-e^{-x^{2}/(2\sigma ^{2})}}

for 
 
 
 
 x
 ∈
 [
 0
 ,
 ∞
 )
 .
 
 
 {\displaystyle x\in [0,\infty ).}

<h2 id="relation-to-random-vector-length">Relation to random vector length</h2>
Consider the two-dimensional vector 
 
 
 
 Y
 =
 (
 U
 ,
 V
 )
 
 
 {\displaystyle Y=(U,V)}
 
 which has components that are <a href="/facts/Bivariate_normal_distribution/2Xfegqz2">bivariate normally distributed</a>, centered at zero, with equal variances 
 
 
 
 
 σ
 
 2
 
 
 
 
 {\displaystyle \sigma ^{2}}
 
, and independent. Then 
 
 
 
 U
 
 
 {\displaystyle U}
 
 and 
 
 
 
 V
 
 
 {\displaystyle V}
 
 have density functions

f
          
            U
          
        
        (
        x
        ;
        σ
        )
        =
        
          f
          
            V
          
        
        (
        x
        ;
        σ
        )
        =
        
          
            
              e
              
                −
                
                  x
                  
                    2
                  
                
                
                  /
                
                (
                2
                
                  σ
                  
                    2
                  
                
                )
              
            
            
              2
              π
              
                σ
                
                  2
                
              
            
          
        
        .
      
    
    {\displaystyle f_{U}(x;\sigma )=f_{V}(x;\sigma )={\frac {e^{-x^{2}/(2\sigma ^{2})}}{\sqrt {2\pi \sigma ^{2}}}}.}

Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be the length of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
. That is, 
 
 
 
 X
 =
 
 
 
 U
 
 2
 
 
 +
 
 V
 
 2
 
 
 
 
 .
 
 
 {\displaystyle X={\sqrt {U^{2}+V^{2}}}.}
 
 Then 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has cumulative distribution function

F
          
            X
          
        
        (
        x
        ;
        σ
        )
        =
        
          ∬
          
            
              D
              
                x
              
            
          
        
        
          f
          
            U
          
        
        (
        u
        ;
        σ
        )
        
          f
          
            V
          
        
        (
        v
        ;
        σ
        )
        
        d
        A
        ,
      
    
    {\displaystyle F_{X}(x;\sigma )=\iint _{D_{x}}f_{U}(u;\sigma )f_{V}(v;\sigma )\,dA,}

where 
 
 
 
 
 D
 
 x
 
 
 
 
 {\displaystyle D_{x}}
 
 is the disk

D
          
            x
          
        
        =
        
          {
          
            (
            u
            ,
            v
            )
            :
            
              
                
                  u
                  
                    2
                  
                
                +
                
                  v
                  
                    2
                  
                
              
            
            ≤
            x
          
          }
        
        .
      
    
    {\displaystyle D_{x}=\left\{(u,v):{\sqrt {u^{2}+v^{2}}}\leq x\right\}.}

Writing the <a href="/facts/Multiple_integral/Fvl3Bkam">double integral</a> in <a href="/facts/Polar_coordinate_system/HBko4YoN">polar coordinates</a>, it becomes

F
          
            X
          
        
        (
        x
        ;
        σ
        )
        =
        
          
            1
            
              2
              π
              
                σ
                
                  2
                
              
            
          
        
        
          ∫
          
            0
          
          
            2
            π
          
        
        
          ∫
          
            0
          
          
            x
          
        
        r
        
          e
          
            −
            
              r
              
                2
              
            
            
              /
            
            (
            2
            
              σ
              
                2
              
            
            )
          
        
        
        d
        r
        
        d
        θ
        =
        
          
            1
            
              σ
              
                2
              
            
          
        
        
          ∫
          
            0
          
          
            x
          
        
        r
        
          e
          
            −
            
              r
              
                2
              
            
            
              /
            
            (
            2
            
              σ
              
                2
              
            
            )
          
        
        
        d
        r
        .
      
    
    {\displaystyle F_{X}(x;\sigma )={\frac {1}{2\pi \sigma ^{2}}}\int _{0}^{2\pi }\int _{0}^{x}re^{-r^{2}/(2\sigma ^{2})}\,dr\,d\theta ={\frac {1}{\sigma ^{2}}}\int _{0}^{x}re^{-r^{2}/(2\sigma ^{2})}\,dr.}

Finally, the probability density function for 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the derivative of its cumulative distribution function, which by the <a href="/facts/Fundamental_theorem_of_calculus/pHeSGtUx">fundamental theorem of calculus</a> is

f
          
            X
          
        
        (
        x
        ;
        σ
        )
        =
        
          
            d
            
              d
              x
            
          
        
        
          F
          
            X
          
        
        (
        x
        ;
        σ
        )
        =
        
          
            x
            
              σ
              
                2
              
            
          
        
        
          e
          
            −
            
              x
              
                2
              
            
            
              /
            
            (
            2
            
              σ
              
                2
              
            
            )
          
        
        ,
      
    
    {\displaystyle f_{X}(x;\sigma )={\frac {d}{dx}}F_{X}(x;\sigma )={\frac {x}{\sigma ^{2}}}e^{-x^{2}/(2\sigma ^{2})},}

which is the Rayleigh distribution. It is straightforward to generalize to vectors of dimension other than 2. 
There are also generalizations when the components have <a href="/facts/Unequal_variance/msBQWI7G">unequal variance</a> or correlations (<a href="/facts/Hoyt_distribution/HoyxF3AF">Hoyt distribution</a>), or when the vector Y follows a <a href="/facts/Multivariate_t-distribution/WspgrIac">bivariate Student t-distribution</a> (see also: <a href="/facts/Hotelling%27s_T-squared_distribution/RJCmedIp">Hotelling's T-squared distribution</a>).<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<table><tbody><tr><th>Generalization to bivariate Student's t-distribution</th></tr><tr><td>Suppose Y {\displaystyle Y} is a random vector with components u , v {\displaystyle u,v} that follows a <a href="/facts/Multivariate_t-distribution/WspgrIac">multivariate t-distribution</a>. If the components both have mean zero, equal variance and are independent, the bivariate Student's-t distribution takes the form: f ( u , v ) = 1 2 π σ 2 ( 1 + u 2 + v 2 ν σ 2 ) − ν / 2 − 1 {\displaystyle f(u,v)={1 \over {2\pi \sigma ^{2}}}\left(1+{u^{2}+v^{2} \over {\nu \sigma ^{2}}}\right)^{-\nu /2-1}} Let R = U 2 + V 2 {\displaystyle R={\sqrt {U^{2}+V^{2}}}} be the magnitude of Y {\displaystyle Y} . Then the <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> (CDF) of the magnitude is: F ( r ) = 1 2 π σ 2 ∬ D r ( 1 + u 2 + v 2 ν σ 2 ) − ν / 2 − 1 d u d v {\displaystyle F(r)={1 \over {2\pi \sigma ^{2}}}\iint _{D_{r}}\left(1+{u^{2}+v^{2} \over {\nu \sigma ^{2}}}\right)^{-\nu /2-1}du\;dv} where D r {\displaystyle D_{r}} is the disk defined by: D r = { ( u , v ) : u 2 + v 2 ≤ r } {\displaystyle D_{r}=\left\{(u,v):{\sqrt {u^{2}+v^{2}}}\leq r\right\}} Converting to <a href="/facts/Polar_coordinates/HBko4YoN">polar coordinates</a> leads to the CDF becoming: F ( r ) = 1 2 π σ 2 ∫ 0 r ∫ 0 2 π ρ ( 1 + ρ 2 ν σ 2 ) − ν / 2 − 1 d θ d ρ = 1 σ 2 ∫ 0 r ρ ( 1 + ρ 2 ν σ 2 ) − ν / 2 − 1 d ρ = 1 − ( 1 + r 2 ν σ 2 ) − ν / 2 {\displaystyle {\begin{aligned}F(r)&={1 \over {2\pi \sigma ^{2}}}\int _{0}^{r}\int _{0}^{2\pi }\rho \left(1+{\rho ^{2} \over {\nu \sigma ^{2}}}\right)^{-\nu /2-1}d\theta \;d\rho \\&={1 \over {\sigma ^{2}}}\int _{0}^{r}\rho \left(1+{\rho ^{2} \over {\nu \sigma ^{2}}}\right)^{-\nu /2-1}d\rho \\&=1-\left(1+{r^{2} \over {\nu \sigma ^{2}}}\right)^{-\nu /2}\end{aligned}}} Finally, the <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> (PDF) of the magnitude may be derived: f ( r ) = F ′ ( r ) = r σ 2 ( 1 + r 2 ν σ 2 ) − ν / 2 − 1 {\displaystyle f(r)=F'(r)={r \over {\sigma ^{2}}}\left(1+{r^{2} \over {\nu \sigma ^{2}}}\right)^{-\nu /2-1}} In the limit as ν → ∞ {\displaystyle \nu \rightarrow \infty } , the Rayleigh distribution is recovered because: lim ν → ∞ ( 1 + r 2 ν σ 2 ) − ν / 2 − 1 = e − r 2 / 2 σ 2 {\displaystyle \lim _{\nu \rightarrow \infty }\left(1+{r^{2} \over {\nu \sigma ^{2}}}\right)^{-\nu /2-1}=e^{-r^{2}/2\sigma ^{2}}} </td></tr></tbody></table>
<h2 id="properties">Properties</h2>
The <a href="/facts/Moment_(mathematics)/GL7vJ1nb">raw moments</a> are given by:

μ
          
            j
          
        
        =
        
          σ
          
            j
          
        
        
          2
          
            j
            
              /
            
            2
          
        
        
        Γ
        
          (
          
            1
            +
            
              
                j
                2
              
            
          
          )
        
        ,
      
    
    {\displaystyle \mu _{j}=\sigma ^{j}2^{j/2}\,\Gamma \left(1+{\frac {j}{2}}\right),}

where 
 
 
 
 Γ
 (
 z
 )
 
 
 {\displaystyle \Gamma (z)}
 
 is the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>.
The <a href="/facts/Mean/swcEd4Pg">mean</a> of a Rayleigh random variable is thus :

μ
        (
        X
        )
        =
        σ
        
          
            
              π
              2
            
          
        
         
        ≈
        1.253
         
        σ
        .
      
    
    {\displaystyle \mu (X)=\sigma {\sqrt {\frac {\pi }{2}}}\ \approx 1.253\ \sigma .}

The <a href="/facts/Standard_deviation/qSug9BRl">standard deviation</a> of a Rayleigh random variable is:

std
        ⁡
        (
        X
        )
        =
        
          
            
              (
              
                2
                −
                
                  
                    π
                    2
                  
                
              
              )
            
          
        
        σ
        ≈
        0.655
         
        σ
      
    
    {\displaystyle \operatorname {std} (X)={\sqrt {\left(2-{\frac {\pi }{2}}\right)}}\sigma \approx 0.655\ \sigma }

The <a href="/facts/Variance/ULBJKXD1">variance</a> of a Rayleigh random variable is :

var
        ⁡
        (
        X
        )
        =
        
          μ
          
            2
          
        
        −
        
          μ
          
            1
          
          
            2
          
        
        =
        
          (
          
            2
            −
            
              
                π
                2
              
            
          
          )
        
        
          σ
          
            2
          
        
        ≈
        0.429
         
        
          σ
          
            2
          
        
      
    
    {\displaystyle \operatorname {var} (X)=\mu _{2}-\mu _{1}^{2}=\left(2-{\frac {\pi }{2}}\right)\sigma ^{2}\approx 0.429\ \sigma ^{2}}

The <a href="/facts/Mode_(statistics)/vq5tGp0G">mode</a> is 
 
 
 
 σ
 ,
 
 
 {\displaystyle \sigma ,}
 
 and the maximum pdf is

f
          
            max
          
        
        =
        f
        (
        σ
        ;
        σ
        )
        =
        
          
            1
            σ
          
        
        
          e
          
            −
            1
            
              /
            
            2
          
        
        ≈
        
          
            0.606
            σ
          
        
        .
      
    
    {\displaystyle f_{\max }=f(\sigma ;\sigma )={\frac {1}{\sigma }}e^{-1/2}\approx {\frac {0.606}{\sigma }}.}

The <a href="/facts/Skewness/rdS8luFa">skewness</a> is given by:

γ
          
            1
          
        
        =
        
          
            
              2
              
                
                  π
                
              
              (
              π
              −
              3
              )
            
            
              (
              4
              −
              π
              
                )
                
                  3
                  
                    /
                  
                  2
                
              
            
          
        
        ≈
        0.631
      
    
    {\displaystyle \gamma _{1}={\frac {2{\sqrt {\pi }}(\pi -3)}{(4-\pi )^{3/2}}}\approx 0.631}

The excess <a href="/facts/Kurtosis/yqGgry89">kurtosis</a> is given by:

γ
          
            2
          
        
        =
        −
        
          
            
              6
              
                π
                
                  2
                
              
              −
              24
              π
              +
              16
            
            
              (
              4
              −
              π
              
                )
                
                  2
                
              
            
          
        
        ≈
        0.245
      
    
    {\displaystyle \gamma _{2}=-{\frac {6\pi ^{2}-24\pi +16}{(4-\pi )^{2}}}\approx 0.245}

The <a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">characteristic function</a> is given by:

φ
        (
        t
        )
        =
        1
        −
        σ
        t
        
          e
          
            −
            
              
                1
                2
              
            
            
              σ
              
                2
              
            
            
              t
              
                2
              
            
          
        
        
          
            
              π
              2
            
          
        
        
          [
          
            erfi
            ⁡
            
              (
              
                
                  
                    σ
                    t
                  
                  
                    2
                  
                
              
              )
            
            −
            i
          
          ]
        
      
    
    {\displaystyle \varphi (t)=1-\sigma te^{-{\frac {1}{2}}\sigma ^{2}t^{2}}{\sqrt {\frac {\pi }{2}}}\left[\operatorname {erfi} \left({\frac {\sigma t}{\sqrt {2}}}\right)-i\right]}

where 
 
 
 
 erfi
 ⁡
 (
 z
 )
 
 
 {\displaystyle \operatorname {erfi} (z)}
 
 is the imaginary <a href="/facts/Error_function/Ca2H7pGr">error function</a>. The <a href="/facts/Moment_generating_function/eTmk43QU">moment generating function</a> is given by

M
        (
        t
        )
        =
        1
        +
        σ
        t
        
        
          e
          
            
              
                1
                2
              
            
            
              σ
              
                2
              
            
            
              t
              
                2
              
            
          
        
        
          
            
              π
              2
            
          
        
        
          [
          
            erf
            ⁡
            
              (
              
                
                  
                    σ
                    t
                  
                  
                    2
                  
                
              
              )
            
            +
            1
          
          ]
        
      
    
    {\displaystyle M(t)=1+\sigma t\,e^{{\frac {1}{2}}\sigma ^{2}t^{2}}{\sqrt {\frac {\pi }{2}}}\left[\operatorname {erf} \left({\frac {\sigma t}{\sqrt {2}}}\right)+1\right]}

where 
 
 
 
 erf
 ⁡
 (
 z
 )
 
 
 {\displaystyle \operatorname {erf} (z)}
 
 is the <a href="/facts/Error_function/Ca2H7pGr">error function</a>.

<h3>Differential entropy</h3>
The <a href="/facts/Differential_entropy/EoJSyw95">differential entropy</a> is given by

H
        =
        1
        +
        ln
        ⁡
        
          (
          
            
              σ
              
                2
              
            
          
          )
        
        +
        
          
            γ
            2
          
        
      
    
    {\displaystyle H=1+\ln \left({\frac {\sigma }{\sqrt {2}}}\right)+{\frac {\gamma }{2}}}

where 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 is the <a href="/facts/Euler%E2%80%93Mascheroni_constant/2kEgRvjC">Euler–Mascheroni constant</a>.

<h2 id="parameter-estimation">Parameter estimation</h2>
Given a sample of N <a href="/facts/Independent_and_identically_distributed/othIRaWt">independent and identically distributed</a> Rayleigh random variables 
 
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x_{i}}
 
 with parameter 
 
 
 
 σ
 
 
 {\displaystyle \sigma }
 
,

σ
 
 2
 
 
 ^
 
 
 
 =
 
 
 
 
 1
 
 2
 N
 
 
 
 
 ∑
 
 i
 =
 1
 
 
 N
 
 
 
 x
 
 i
 
 
 2
 
 
 
 
 {\displaystyle {\widehat {\sigma ^{2}}}=\!\,{\frac {1}{2N}}\sum _{i=1}^{N}x_{i}^{2}}
 
 is the <a href="/facts/Maximum_likelihood_estimation/0Yq2dpQD">maximum likelihood</a> estimate and also is <a href="/facts/Bias_of_an_estimator/oxIvEgmd">unbiased</a>.

σ
              ^
            
          
        
        ≈
        
          
            
              
                1
                
                  2
                  N
                
              
            
            
              ∑
              
                i
                =
                1
              
              
                N
              
            
            
              x
              
                i
              
              
                2
              
            
          
        
      
    
    {\displaystyle {\widehat {\sigma }}\approx {\sqrt {{\frac {1}{2N}}\sum _{i=1}^{N}x_{i}^{2}}}}
  
 is a biased estimator that can be corrected via the formula

σ
 =
 
 
 
 σ
 ^
 
 
 
 
 
 
 Γ
 (
 N
 )
 
 
 N
 
 
 
 
 Γ
 
 (
 
 N
 +
 
 
 1
 2
 
 
 
 )
 
 
 
 
 =
 
 
 
 σ
 ^
 
 
 
 
 
 
 
 4
 
 N
 
 
 N
 !
 (
 N
 −
 1
 )
 !
 
 
 N
 
 
 
 
 (
 2
 N
 )
 !
 
 
 π
 
 
 
 
 
 
 
 {\displaystyle \sigma ={\widehat {\sigma }}{\frac {\Gamma (N){\sqrt {N}}}{\Gamma \left(N+{\frac {1}{2}}\right)}}={\widehat {\sigma }}{\frac {4^{N}N!(N-1)!{\sqrt {N}}}{(2N)!{\sqrt {\pi }}}}}
 
<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> 
 
 
 
 =
 
 
 
 
 σ
 ^
 
 
 
 
 c
 
 4
 
 
 (
 2
 N
 +
 1
 )
 
 
 
 
 
 {\displaystyle ={\frac {\hat {\sigma }}{c_{4}(2N+1)}}}
 
, where <a href="/facts/Standard_deviation/qSug9BRl">c4 is the correction factor used to unbias estimates of standard deviation for normal random variables</a>.
<h3>Confidence intervals</h3>
To find the (1 − α) confidence interval, first find the bounds 
 
 
 
 [
 a
 ,
 b
 ]
 
 
 {\displaystyle [a,b]}
 
 where:

  
 
 
 
 P
 
 (
 
 
 χ
 
 2
 N
 
 
 2
 
 
 ≤
 a
 
 )
 
 =
 α
 
 /
 
 2
 ,
 
 P
 
 (
 
 
 χ
 
 2
 N
 
 
 2
 
 
 ≤
 b
 
 )
 
 =
 1
 −
 α
 
 /
 
 2
 
 
 {\displaystyle P\left(\chi _{2N}^{2}\leq a\right)=\alpha /2,\quad P\left(\chi _{2N}^{2}\leq b\right)=1-\alpha /2}

then the scale parameter will fall within the bounds

  
 
 
 
 
 
 
 
 N
 
 
 
 
 x
 
 2
 
 
 ¯
 
 
 
 b
 
 
 ≤
 
 
 
 
 σ
 
 2
 
 
 ^
 
 
 
 ≤
 
 
 
 
 N
 
 
 
 
 x
 
 2
 
 
 ¯
 
 
 
 a
 
 
 
 
 {\displaystyle {\frac {{N}{\overline {x^{2}}}}{b}}\leq {\widehat {\sigma ^{2}}}\leq {\frac {{N}{\overline {x^{2}}}}{a}}}
 
<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
<h2 id="generating-random-variates">Generating random variates</h2>
Given a random variate U drawn from the <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT">uniform distribution</a> in the interval (0, 1), then the variate

X
        =
        σ
        
          
            −
            2
            ln
            ⁡
            U
          
        
        
      
    
    {\displaystyle X=\sigma {\sqrt {-2\ln U}}\,}

has a Rayleigh distribution with parameter 
 
 
 
 σ
 
 
 {\displaystyle \sigma }
 
. This is obtained by applying the <a href="/facts/Inverse_transform_sampling/fIDxBxT6">inverse transform sampling</a>-method.

<h2 id="related-distributions">Related distributions</h2>
<ul><li>
 
 
 
 R
 ∼
 
 R
 a
 y
 l
 e
 i
 g
 h
 
 (
 σ
 )
 
 
 {\displaystyle R\sim \mathrm {Rayleigh} (\sigma )}
 
 is Rayleigh distributed if 
 
 
 
 R
 =
 
 
 
 X
 
 2
 
 
 +
 
 Y
 
 2
 
 
 
 
 
 
 {\displaystyle R={\sqrt {X^{2}+Y^{2}}}}
 
, where 
 
 
 
 X
 ∼
 N
 (
 0
 ,
 
 σ
 
 2
 
 
 )
 
 
 {\displaystyle X\sim N(0,\sigma ^{2})}
 
 and 
 
 
 
 Y
 ∼
 N
 (
 0
 ,
 
 σ
 
 2
 
 
 )
 
 
 {\displaystyle Y\sim N(0,\sigma ^{2})}
 
 are independent <a href="/facts/Normal_distribution/UapjjPyQ">normal random variables</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> This gives motivation to the use of the symbol 
 
 
 
 σ
 
 
 {\displaystyle \sigma }
 
 in the above parametrization of the Rayleigh density.</li>
<li>The magnitude 
 
 
 
 
 |
 
 z
 
 |
 
 
 
 {\displaystyle |z|}
 
 of a <a href="/facts/Standard_complex_normal_distribution/KVUmoU1L">standard complex normally distributed</a> variable z is Rayleigh distributed.</li>
<li>The <a href="/facts/Chi_distribution/bV4MEanU">chi distribution</a> with v = 2 is equivalent to the Rayleigh Distribution with σ = 1: 
 
 
 
 R
 (
 σ
 )
 ∼
 σ
 
 χ
 
 2
 
 
 
 
 
  
 .
 
 
 {\displaystyle R(\sigma )\sim \sigma \chi _{2}^{\,}\ .}
 
</li>
<li>If 
 
 
 
 R
 ∼
 
 R
 a
 y
 l
 e
 i
 g
 h
 
 (
 1
 )
 
 
 {\displaystyle R\sim \mathrm {Rayleigh} (1)}
 
, then 
 
 
 
 
 R
 
 2
 
 
 
 
 {\displaystyle R^{2}}
 
 has a <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared distribution</a> with 2 degrees of freedom: 
 
 
 
 [
 Q
 =
 R
 (
 σ
 
 )
 
 2
 
 
 ]
 ∼
 
 σ
 
 2
 
 
 
 χ
 
 2
 
 
 2
 
 
  
 .
 
 
 {\displaystyle [Q=R(\sigma )^{2}]\sim \sigma ^{2}\chi _{2}^{2}\ .}
 
</li>
<li>If 
 
 
 
 R
 ∼
 
 R
 a
 y
 l
 e
 i
 g
 h
 
 (
 σ
 )
 
 
 {\displaystyle R\sim \mathrm {Rayleigh} (\sigma )}
 
, then 
 
 
 
 
 ∑
 
 i
 =
 1
 
 
 N
 
 
 
 R
 
 i
 
 
 2
 
 
 
 
 {\displaystyle \sum _{i=1}^{N}R_{i}^{2}}
 
 has a <a href="/facts/Gamma_distribution/lczcdJmw">gamma distribution</a> with integer scale parameter 
 
 
 
 N
 
 
 {\displaystyle N}
 
 and rate parameter 
 
 
 
 
 
 1
 
 2
 
 σ
 
 2
 
 
 
 
 
 
 
 {\displaystyle {\frac {1}{2\sigma ^{2}}}}

[
 
 Y
 =
 
 ∑
 
 i
 =
 1
 
 
 N
 
 
 
 R
 
 i
 
 
 2
 
 
 
 ]
 
 ∼
 Γ
 
 (
 
 N
 ,
 
 
 1
 
 2
 
 σ
 
 2
 
 
 
 
 
 
 )
 
 
 
 {\displaystyle \left[Y=\sum _{i=1}^{N}R_{i}^{2}\right]\sim \Gamma \left(N,{\frac {1}{2\sigma ^{2}}}\right)}
 
 with integer shape parameter N and rate parameter 
 
 
 
 
 
 1
 
 2
 
 σ
 
 2
 
 
 
 
 
 .
 
 
 {\displaystyle {\frac {1}{2\sigma ^{2}}}.}

[
 
 Y
 =
 
 ∑
 
 i
 =
 1
 
 
 N
 
 
 
 R
 
 i
 
 
 2
 
 
 
 ]
 
 ∼
 Γ
 
 (
 
 N
 ,
 2
 
 σ
 
 2
 
 
 
 )
 
 
 
 {\displaystyle \left[Y=\sum _{i=1}^{N}R_{i}^{2}\right]\sim \Gamma \left(N,2\sigma ^{2}\right)}
 
 with integer shape parameter N and scale parameter 
 
 
 
 2
 
 σ
 
 2
 
 
 .
 
 
 {\displaystyle 2\sigma ^{2}.}
 
</li>
<li>The <a href="/facts/Rice_distribution/E9tY1XNJ">Rice distribution</a> is a <a href="/facts/Noncentral_distribution/uWgoRYTm">noncentral generalization</a> of the Rayleigh distribution: 
 
 
 
 
 R
 a
 y
 l
 e
 i
 g
 h
 
 (
 σ
 )
 =
 
 R
 i
 c
 e
 
 (
 0
 ,
 σ
 )
 
 
 {\displaystyle \mathrm {Rayleigh} (\sigma )=\mathrm {Rice} (0,\sigma )}
 
.</li>
<li>The <a href="/facts/Weibull_distribution/1fEWxbKl">Weibull distribution</a> with the <a href="/facts/Shape_parameter/7jaG8RDp">shape parameter</a> k = 2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter 
 
 
 
 σ
 
 
 {\displaystyle \sigma }
 
 is related to the Weibull scale parameter according to 
 
 
 
 λ
 =
 σ
 
 
 2
 
 
 .
 
 
 {\displaystyle \lambda =\sigma {\sqrt {2}}.}
 
</li>
<li>If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has an <a href="/facts/Exponential_distribution/yY3EdV0u">exponential distribution</a> 
 
 
 
 X
 ∼
 
 E
 x
 p
 o
 n
 e
 n
 t
 i
 a
 l
 
 (
 λ
 )
 
 
 {\displaystyle X\sim \mathrm {Exponential} (\lambda )}
 
, then 
 
 
 
 Y
 =
 
 
 X
 
 
 ∼
 
 R
 a
 y
 l
 e
 i
 g
 h
 
 (
 1
 
 /
 
 
 
 2
 λ
 
 
 )
 .
 
 
 {\displaystyle Y={\sqrt {X}}\sim \mathrm {Rayleigh} (1/{\sqrt {2\lambda }}).}
 
</li>
<li>The <a href="/facts/Half-normal_distribution/Ed15Xhlj">half-normal distribution</a> is the one-dimensional equivalent of the Rayleigh distribution.</li>
<li>The <a href="/facts/Maxwell%E2%80%93Boltzmann_distribution/KAuVfdKJ">Maxwell–Boltzmann distribution</a> is the three-dimensional equivalent of the Rayleigh distribution.</li></ul>
<h2 id="applications">Applications</h2>
An application of the estimation of σ can be found in <a href="/facts/Magnetic_resonance_imaging/Vh9yG9qq">magnetic resonance imaging</a> (MRI). As MRI images are recorded as <a href="/facts/Complex_numbers/2w7mqI7e">complex</a> images but most often viewed as magnitude images, the background data is Rayleigh distributed. Hence, the above formula can be used to estimate the noise variance in an MRI image from background data.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> 
The Rayleigh distribution was also employed in the field of <a href="/facts/Nutrition/SMJ3Dhfs">nutrition</a> for linking <a href="/facts/Diet_(nutrition)/7qzDHpop">dietary</a> <a href="/facts/Nutrient/34bn4skK">nutrient</a> levels and <a href="/facts/Human/rlcTENsr">human</a> and <a href="/facts/Animal_husbandry/aJforlQp">animal</a> responses. In this way, the <a href="/facts/Parameter/zFvVFMv9">parameter</a> σ may be used to calculate nutrient response relationship.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>
In the field of <a href="/facts/Ballistics/3WHTox0u">ballistics</a>, the Rayleigh distribution is used for calculating the <a href="/facts/Circular_error_probable/KF8Shk8G">circular error probable</a>—a measure of a gun's precision.
In <a href="/facts/Physical_oceanography/E2gVWg8N">physical oceanography</a>, the distribution of <a href="/facts/Significant_wave_height/a20bxsSL">significant wave height</a> approximately follows a Rayleigh distribution.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Circular_error_probable/KF8Shk8G">Circular error probable</a></li>
<li><a href="/facts/Rayleigh_fading/ryY9Bm9I">Rayleigh fading</a></li>
<li><a href="/facts/Rayleigh_mixture_distribution/eSRhmQdo">Rayleigh mixture distribution</a></li>
<li><a href="/facts/Rice_distribution/E9tY1XNJ">Rice distribution</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"The Wave Theory of Light", Encyclopedic Britannica 1888; "The Problem of the Random Walk", Nature 1905 vol.72 p.318 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Papoulis, Athanasios; Pillai, S. (2001) Probability, Random Variables and Stochastic Processes. ISBN 0073660116, ISBN 9780073660110 [page needed] <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Papoulis, Athanasios; Pillai, S. (2001) Probability, Random Variables and Stochastic Processes. ISBN 0073660116, ISBN 9780073660110 [page needed] <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Röver, C. (2011). "Student-t based filter for robust signal detection". Physical Review D. 84 (12): 122004. arXiv:1109.0442. Bibcode:2011PhRvD..84l2004R. doi:10.1103/physrevd.84.122004. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Siddiqui, M. M. (1964) "Statistical inference for Rayleigh distributions", The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9, p. 1007 <a href="https://archive.org/details/jresv68Dn9p1005" target="_blank">https://archive.org/details/jresv68Dn9p1005</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Siddiqui, M. M. (1961) "Some Problems Connected With Rayleigh Distributions", The Journal of Research of the National Bureau of Standards; Sec. D: Radio Propagation, Vol. 66D, No. 2, p. 169 <a href="http://nvlpubs.nist.gov/nistpubs/jres/66D/jresv66Dn2p167_A1b.pdf" target="_blank">http://nvlpubs.nist.gov/nistpubs/jres/66D/jresv66Dn2p167_A1b.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Hogema, Jeroen (2005) "Shot group statistics" <a href="https://web.archive.org/web/20131105232146/http://home.kpn.nl/jhhogema1966/skeetn/ballist/sgs/sgs.htm#_Toc96439743" target="_blank">https://web.archive.org/web/20131105232146/http://home.kpn.nl/jhhogema1966/skeetn/ballist/sgs/sgs.htm#_Toc96439743</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Sijbers, J.; den Dekker, A. J.; Raman, E.; Van Dyck, D. (1999). "Parameter estimation from magnitude MR images". International Journal of Imaging Systems and Technology. 10 (2): 109–114. CiteSeerX 10.1.1.18.1228. doi:10.1002/(sici)1098-1098(1999)10:2<109::aid-ima2>3.0.co;2-r. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">den Dekker, A. J.; Sijbers, J. (2014). "Data distributions in magnetic resonance images: a review". Physica Medica. 30 (7): 725–741. doi:10.1016/j.ejmp.2014.05.002. PMID 25059432. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Ahmadi, Hamed (2017-11-21). "A mathematical function for the description of nutrient-response curve". PLOS ONE. 12 (11): e0187292. Bibcode:2017PLoSO..1287292A. doi:10.1371/journal.pone.0187292. ISSN 1932-6203. PMC 5697816. PMID 29161271. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5697816" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5697816</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">"Rayleigh Probability Distribution Applied to Random Wave Heights" (PDF). United States Naval Academy. <a href="https://www.usna.edu/NAOE/_files/documents/Courses/EN330/Rayleigh-Probability-Distribution-Applied-to-Random-Wave-Heights.pdf" target="_blank">https://www.usna.edu/NAOE/_files/documents/Courses/EN330/Rayleigh-Probability-Distribution-Applied-to-Random-Wave-Heights.pdf</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
</ol>

Rayleigh distribution open-in-new

Rayleigh distribution